Answer the question below based on the given information.

Two containers of solutions, A and B, contain alcohol and water in the ratio of 5 : 4 and 2 : 1 respectively. A portion equal to 20% of solution A and 50% of solution B is combined to create a third solution C. If the total quantity of alcohol present in solution C is 90 liters and the ratio of alcohol to water in solution C is 30 : 19, then determine the initial quantity of solution A.

The correct option is 360 liters.

Let's solve the problem step-by-step to find the initial quantity of solution A.

Step 1: Understand the ratio of alcohol and water in solutions A and B

For Solution A, the ratio of alcohol to water is 5 : 4.
This means in Solution A:
Fraction of alcohol = 5 / (5 + 4) = 5/9
Fraction of water = 4 / (5 + 4) = 4/9

For Solution B, the ratio of alcohol to water is 2 : 1.
This means in Solution B:
Fraction of alcohol = 2 / (2 + 1) = 2/3
Fraction of water = 1 / (2 + 1) = 1/3

Step 2: Express the quantities taken from A and B

Let the total initial volume of Solution A be $A$ liters and Solution B be $B$ liters.

According to the question:
20% of solution A is taken = 0.20$A$ = (1/5)$A$
50% of solution B is taken = 0.50$B$ = (1/2)$B$

Step 3: Calculate the total alcohol in Solution C

The alcohol coming from Solution A is:

$\text{Alcohol from A}=\frac{5}{9}\×\frac{1}{5}A=\frac{1}{9}A$

The alcohol coming from Solution B is:

$\text{Alcohol from B}=\frac{2}{3}\×\frac{1}{2}B=\frac{1}{3}B$

The total alcohol in Solution C is given as 90 liters. Therefore:

$\frac{1}{9}A+\frac{1}{3}B=90$

Multiplying the entire equation by 9 to clear fractions:

$A+3B=810$         (Equation 1)

Step 4: Calculate the total water in Solution C

The ratio of alcohol to water in Solution C is 30 : 19.
Since total alcohol in C is 90 liters, let's find the total water in C:

$\frac{\text{Alcohol in C}}{\text{Water in C}}=\frac{30}{19}$

$\frac{90}{\text{Water in C}}=\frac{30}{19}$

$\text{Water in C}=\frac{90\×19}{30}=3\×19=57\text{liters}$

Now, let's express the water in C in terms of $A$ and $B$:
Water from Solution A = (4/9) × (1/5)$A$ = (4/45)$A$
Water from Solution B = (1/3) × (1/2)$B$ = (1/6)$B$

So the total water equation is:

$\frac{4}{45}A+\frac{1}{6}B=57$

Multiply the entire equation by 90 to clear fractions:

$8A+15B=5130$         (Equation 2)

Step 5: Solve the equations to find $A$

From Equation 1, express $B$ in terms of $A$:

$3B=810-A$

Multiply by 5:

$15B=4050-5A$

Substitute $15B$ into Equation 2:

$8A+(4050-5A)=5130$

$3A+4050=5130$

$3A=5130-4050$

$3A=1080$

$A=\frac{1080}{3}=360\text{liters}$

Thus, the initial quantity of solution A is 360 liters.